Regarding the proof of Theorem REF , the key complication is that the spin\(^c\)  structures \(\mathfrak {s}_L\)  are torsion. This implies that even though it is true for a general automorphism \(\varphi \)  of a Riemann surface that the equations of \(M_{\varphi }\)  do not admit irreducible solutions for specific metrics and perturbations [1]},[2]}, the unperturbed Seiberg-Witten equations always admit a \(b_1(M_{\varphi })\) -dimensional torus \(\mathbb {T}\)  of reducible solutions [3]}, which turns out to be highly degenerate: for example, in the case \(\varphi \)  is the identity, the Hessian of the Chern-Simons-Dirac functional is singular along a copy of the (\(2g-2\) )-dimensional theta divisor of \(\Sigma \)  in the (\(2g+1\) )-dimensional torus of reducible solutions on \(M_{\varphi }=S^1\times \Sigma \) ; the computation of its Floer homology via surgery techniques [4]} indeed confirms that very delicate things must happen when trying to perturb the equations to achieve transversality.
